Asymptotics in Statistics
Some Basic Concepts
1st Edition
Published on 6. December 2012
VIII, 180 pages
978-1-4684-0377-0 (ISBN)
Description
In the summer of 1968 one of the present authors (LLC) had the pleasure of giving a sequence of lectures at the University of Mon treal. Lecture notes were collected and written out by Drs. Catherine Doleans, Jean Haezendonck and Roch Roy. They were published in French by the Presses of the University of Montreal as part of their series of Seminaires de Mathematiques Superieures. Twenty years later it was decided that a Chinese translation could be useful, but upon prodding by Professor Shanti Gupta at Purdue we concluded that the notes should be updated and rewritten in English and in Chinese. The present volume is the result of that effort. We have preserved the general outline of the lecture notes, but we have deleted obsolete material and sketched some of the results acquired during the past twenty years. This means that while the original notes concentrated on the LAN situation we have included here some results of Jeganathan and others on the LAMN case. Also included are versions of the Hajek-Le Cam asymptotic minimax and convolution theorems with some of their implications. We have not attempted to give complete coverage of the subject and have often stated theorems without indicating their proofs.
Reviews / Votes
"Le Cam can be justifiably considered as one of the founders of modern asymptotic theory of statistical inference... the readers will surely appreciate the accessibility of results in a fairly easily understandable setting... Any serious student of asymptotic theory of statistical inference should have Äthis bookÜ on his or her shelf."(Mathtematical Reviews)
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Content
1 Introduction.- 2 Experiments, Deficiencies, Distances.- 2.1 Comparing risk functions.- 2.2 Likelihood ratios. Blackwell's representation.- 2.3 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger distances, Hellinger transforms.- 3.3 Historical Remarks.- 4 Limit Laws for Likelihood Ratios Obtained from Independent Observations.- 4.1 Introduction.- 4.2 Limit laws in the binary case.- 4.3 Historical Remarks.- 5 Locally Asymptotically Normal Families.- 5.1 1 Introduction.- 5.2 Locally asymptotically quadratic families.- 5.3 A method of construction of estimates.- 5.4 Some local Bayes properties.- 5.5 Invariance and regularity.- 5.6 The LAMN and LAN conditions.- 5.7 Additional remarks on the LAN conditions.- 5.8 Wald's tests and confidence ellipsoids.- 5.9 Possible extensions.- 5.10 Historical Remarks.- 6 Independent, Identically Distributed Observations.- 6.1 Introduction.- 6.2 The standard i.i.d. case. Differentiability in quadratic mean.- 6.3 Some examples.- 6.4 Some nonparametric considerations.- 6.5 Bounds on the risk of estimates.- 6.6 Some cases where the number of observations is random.- 6.7 Historical Remarks.- 7 On Bayes Procedures.- 7.1 Introduction.- 7.2 Bayes procedures behave nicely.- 7.3 The Bernstein-von Mises Phenomenon.- 7.4 A Bernstein-von Mises result for the i.i.d. case.- 7.5 Bayes procedures behave miserably.- 7.6 Historical Remarks.- 8 Bibliography.- Author Index.
